$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The position of a joint in each rigid body $i$ is given in (7.9-1). The relative motion across a joint is given by the relative translation $$\begin{equation} \mathbf{x}=\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\,,\label{eq:rjm-relative-translation} \end{equation}$$ and the relative rotation $$\begin{equation} \mathbf{Q}=\boldsymbol{\Lambda}^{\left(2\right)}\cdot\left(\boldsymbol{\Lambda}^{\left(1\right)}\right)^{T}\equiv\exp\left[\hat{\boldsymbol{\theta}}\right]\,,\label{eq:rjm-relative-rotation} \end{equation}$$ where $\hat{\boldsymbol{\theta}}=-\boldsymbol{\mathcal{E}}\cdot\boldsymbol{\theta}$ is the skew-symmetric tensor with dual vector $\boldsymbol{\theta}$ . As usual, $\boldsymbol{\theta}$ is a vector whose direction represents the axis of rotation and whose magnitude is the angle of (counter-clockwise) rotation about that axis. To report the relative motion of a joint, we may project $\mathbf{x}$ and $\boldsymbol{\theta}$ along the basis vectors $\mathbf{e}_{j}^{\left(1\right)}=\boldsymbol{\Lambda}^{\left(1\right)}\cdot\mathbf{E}_{j}$ of the first rigid body. Thus, $$\begin{equation} \begin{aligned}x_{j} & =\mathbf{x}\cdot\mathbf{e}_{j}^{\left(1\right)}\\ \theta_{j} & =\boldsymbol{\theta}\cdot\mathbf{e}_{j}^{\left(1\right)} \end{aligned} \,.\label{eq:rjm-projected-motion} \end{equation}$$